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Theorem List for Metamath Proof Explorer - 37901-38000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoreme23an 37901 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜏   )    &   ((𝜒𝜏) → 𝜂)       (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )
 
Theoremee23an 37902 e23an 37901 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓 → (𝜃𝜏)))    &   ((𝜒𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜃𝜂)))
 
Theoreme32 37903 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   (   𝜑   ,   𝜓   ▶   𝜏   )    &   (𝜃 → (𝜏𝜂))       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
 
Theoremee32 37904 e32 37903 without virtual deductions. (Contributed by Alan Sare, 18-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓𝜏))    &   (𝜃 → (𝜏𝜂))       (𝜑 → (𝜓 → (𝜒𝜂)))
 
Theoreme32an 37905 A virtual deduction elimination rule. (Contributed by Alan Sare, 24-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   (   𝜑   ,   𝜓   ▶   𝜏   )    &   ((𝜃𝜏) → 𝜂)       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )
 
Theoremee32an 37906 e33an 37880 without virtual deductions. (Contributed by Alan Sare, 14-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜃)))    &   (𝜑 → (𝜓𝜏))    &   ((𝜃𝜏) → 𝜂)       (𝜑 → (𝜓 → (𝜒𝜂)))
 
Theoreme123 37907 A virtual deduction elimination rule. (Contributed by Alan Sare, 12-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜑   ,   𝜒   ▶   𝜃   )    &   (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜂   )    &   (𝜓 → (𝜃 → (𝜂𝜁)))       (   𝜑   ,   𝜒   ,   𝜏   ▶   𝜁   )
 
Theoremee123 37908 e123 37907 without virtual deductions. (Contributed by Alan Sare, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜑 → (𝜒𝜃))    &   (𝜑 → (𝜒 → (𝜏𝜂)))    &   (𝜓 → (𝜃 → (𝜂𝜁)))       (𝜑 → (𝜒 → (𝜏𝜁)))
 
Theoremel123 37909 A virtual deduction elimination rule. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ▶   𝜓   )    &   (   𝜒   ▶   𝜃   )    &   (   𝜏   ▶   𝜂   )    &   ((𝜓𝜃𝜂) → 𝜁)       (   (   𝜑   ,   𝜒   ,   𝜏   )   ▶   𝜁   )
 
Theoreme233 37910 A virtual deduction elimination rule. (Contributed by Alan Sare, 29-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ▶   𝜒   )    &   (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜏   )    &   (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜂   )    &   (𝜒 → (𝜏 → (𝜂𝜁)))       (   𝜑   ,   𝜓   ,   𝜃   ▶   𝜁   )
 
Theoreme323 37911 A virtual deduction elimination rule. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(   𝜑   ,   𝜓   ,   𝜒   ▶   𝜃   )    &   (   𝜑   ,   𝜓   ▶   𝜏   )    &   (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜂   )    &   (𝜃 → (𝜏 → (𝜂𝜁)))       (   𝜑   ,   𝜓   ,   𝜒   ▶   𝜁   )
 
Theoreme000 37912 A virtual deduction elimination rule. The non-virtual deduction form of e000 37912 is the virtual deduction form. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   𝜒    &   (𝜑 → (𝜓 → (𝜒𝜃)))       𝜃
 
Theoreme00 37913 Elimination rule identical to mp2 9. The non-virtual deduction form is the virtual deduction form, which is mp2 9. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   (𝜑 → (𝜓𝜒))       𝜒
 
Theoreme00an 37914 Elimination rule identical to mp2an 703. The non-virtual deduction form is the virtual deduction form, which is mp2an 703. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒
 
Theoremeel00cT 37915 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   𝜓    &   ((𝜑𝜓) → 𝜒)       (⊤ → 𝜒)
 
TheoremeelTT 37916 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (⊤ → 𝜓)    &   ((𝜑𝜓) → 𝜒)       𝜒
 
Theoreme0a 37917 Elimination rule identical to ax-mp 5. The non-virtual deduction form is the virtual deduction form, which is ax-mp 5. (Contributed by Alan Sare, 14-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       𝜓
 
TheoremeelT 37918 An elimination deduction. (Contributed by Alan Sare, 5-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (𝜑𝜓)       𝜓
 
Theoremeel0cT 37919 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       (⊤ → 𝜓)
 
TheoremeelT0 37920 An elimination deduction. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   𝜓    &   ((𝜑𝜓) → 𝜒)       𝜒
 
Theoreme0bi 37921 Elimination rule identical to mpbi 218. The non-virtual deduction form is the virtual deduction form, which is mpbi 218. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜑𝜓)       𝜓
 
Theoreme0bir 37922 Elimination rule identical to mpbir 219. The non-virtual deduction form is the virtual deduction form, which is mpbir 219. (Contributed by Alan Sare, 15-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜑    &   (𝜓𝜑)       𝜓
 
Theoremuun0.1 37923 Convention notation form of un0.1 37924. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(⊤ → 𝜑)    &   (𝜓𝜒)    &   ((⊤ ∧ 𝜓) → 𝜃)       (𝜓𝜃)
 
Theoremun0.1 37924 is the constant true, a tautology (see df-tru 1477). Kleene's "empty conjunction" is logically equivalent to . In a virtual deduction we shall interpret to be the empty wff or the empty collection of virtual hypotheses. in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(      ▶   𝜑   )    &   (   𝜓   ▶   𝜒   )    &   (   (      ,   𝜓   )   ▶   𝜃   )       (   𝜓   ▶   𝜃   )
 
TheoremuunT1 37925 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) Proof was revised to accomodate a possible future version of df-tru 1477. (Revised by David A. Wheeler, 8-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ 𝜑) → 𝜓)       (𝜑𝜓)
 
TheoremuunT1p1 37926 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∧ ⊤) → 𝜓)       (𝜑𝜓)
 
TheoremuunT21 37927 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ (𝜑𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremuun121 37928 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∧ (𝜑𝜓)) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremuun121p1 37929 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) ∧ 𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremuun132 37930 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∧ (𝜓𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremuun132p1 37931 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜓𝜒) ∧ 𝜑) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremanabss7p1 37932 A deduction unionizing a non-unionized collection of virtual hypotheses. This would have been named uun221 if the 0th permutation did not exist in set.mm as anabss7 857. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜓𝜑) ∧ 𝜑) → 𝜒)       ((𝜓𝜑) → 𝜒)
 
Theoremun10 37933 A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (   𝜑   ,      )   ▶   𝜓   )       (   𝜑   ▶   𝜓   )
 
Theoremun01 37934 A unionizing deduction. (Contributed by Alan Sare, 28-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(   (      ,   𝜑   )   ▶   𝜓   )       (   𝜑   ▶   𝜓   )
 
Theoremun2122 37935 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) ∧ 𝜓𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremuun2131 37936 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜓) ∧ (𝜑𝜒)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremuun2131p1 37937 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜑𝜒) ∧ (𝜑𝜓)) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
TheoremuunTT1 37938 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ ⊤ ∧ 𝜑) → 𝜓)       (𝜑𝜓)
 
TheoremuunTT1p1 37939 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ 𝜑 ∧ ⊤) → 𝜓)       (𝜑𝜓)
 
TheoremuunTT1p2 37940 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∧ ⊤ ∧ ⊤) → 𝜓)       (𝜑𝜓)
 
TheoremuunT11 37941 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ 𝜑𝜑) → 𝜓)       (𝜑𝜓)
 
TheoremuunT11p1 37942 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∧ ⊤ ∧ 𝜑) → 𝜓)       (𝜑𝜓)
 
TheoremuunT11p2 37943 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜑 ∧ ⊤) → 𝜓)       (𝜑𝜓)
 
TheoremuunT12 37944 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ 𝜑𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
TheoremuunT12p1 37945 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((⊤ ∧ 𝜓𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
TheoremuunT12p2 37946 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∧ ⊤ ∧ 𝜓) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
TheoremuunT12p3 37947 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜓 ∧ ⊤ ∧ 𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
TheoremuunT12p4 37948 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓 ∧ ⊤) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
TheoremuunT12p5 37949 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜓𝜑 ∧ ⊤) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremuun111 37950 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜑𝜑) → 𝜓)       (𝜑𝜓)
 
Theorem3anidm12p1 37951 A deduction unionizing a non-unionized collection of virtual hypotheses. 3anidm12 1374 denotes the deduction which would have been named uun112 if it did not pre-exist in set.mm. This second permutation's name is based on this pre-existing name. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜓𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theorem3anidm12p2 37952 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜓𝜑𝜑) → 𝜒)       ((𝜑𝜓) → 𝜒)
 
Theoremuun123 37953 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜒𝜓) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremuun123p1 37954 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜓𝜑𝜒) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremuun123p2 37955 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜒𝜑𝜓) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremuun123p3 37956 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜓𝜒𝜑) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremuun123p4 37957 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜒𝜓𝜑) → 𝜃)       ((𝜑𝜓𝜒) → 𝜃)
 
Theoremuun2221 37958 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 30-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑𝜑 ∧ (𝜓𝜑)) → 𝜒)       ((𝜓𝜑) → 𝜒)
 
Theoremuun2221p1 37959 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜑 ∧ (𝜓𝜑) ∧ 𝜑) → 𝜒)       ((𝜓𝜑) → 𝜒)
 
Theoremuun2221p2 37960 A deduction unionizing a non-unionized collection of virtual hypotheses. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜓𝜑) ∧ 𝜑𝜑) → 𝜒)       ((𝜓𝜑) → 𝜒)
 
Theorem3impdirp1 37961 A deduction unionizing a non-unionized collection of virtual hypotheses. Commuted version of 3impdir 1373. (Contributed by Alan Sare, 4-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(((𝜒𝜓) ∧ (𝜑𝜓)) → 𝜃)       ((𝜑𝜒𝜓) → 𝜃)
 
Theorem3impcombi 37962 A 1-hypothesis propositional calculus deduction. (Contributed by Alan Sare, 25-Sep-2017.)
((𝜑𝜓𝜑) → (𝜒𝜃))       ((𝜓𝜑𝜒) → 𝜃)
 
Theorem3imp231 37963 Importation inference. (Contributed by Alan Sare, 17-Oct-2017.)
(𝜑 → (𝜓 → (𝜒𝜃)))       ((𝜓𝜒𝜑) → 𝜃)
 
20.30.6  Theorems proved using Virtual Deduction
 
TheoremtrsspwALT 37964 Virtual deduction proof of the left-to-right implication of dftr4 4583. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4583 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
 
TheoremtrsspwALT2 37965 Virtual deduction proof of trsspwALT 37964. This proof is the same as the proof of trsspwALT 37964 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A transitive class is a subset of its power class. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
 
TheoremtrsspwALT3 37966 Short predicate calculus proof of the left-to-right implication of dftr4 4583. A transitive class is a subset of its power class. This proof was constructed by applying Metamath's minimize command to the proof of trsspwALT2 37965, which is the virtual deduction proof trsspwALT 37964 without virtual deductions. (Contributed by Alan Sare, 30-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
 
Theoremsspwtr 37967 Virtual deduction proof of the right-to-left implication of dftr4 4583. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 37967 without accumulating results. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)
 
TheoremsspwtrALT 37968 Virtual deduction proof of sspwtr 37967. This proof is the same as the proof of sspwtr 37967 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A class which is a subclass of its power class is transitive. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)
 
TheoremcsbabgOLD 37969* Move substitution into a class abstraction. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) Obsolete as of 19-Aug-2018. Use csbab 3863 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉𝐴 / 𝑥{𝑦𝜑} = {𝑦[𝐴 / 𝑥]𝜑})
 
TheoremcsbunigOLD 37970 Distribute proper substitution through the union of a class. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 22-Aug-2018. Use csbuni 4300 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
 
Theoremcsbfv12gALTOLD 37971 Move class substitution in and out of a function value. The proof is derived from the virtual deduction proof csbfv12gALTVD 38054. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 20-Aug-2018. Use csbfv12 6025 instead. (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
 
TheoremcsbxpgOLD 37972 Distribute proper substitution through the Cartesian product of two classes. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbrn 5404 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝐷𝐴 / 𝑥(𝐵 × 𝐶) = (𝐴 / 𝑥𝐵 × 𝐴 / 𝑥𝐶))
 
TheoremcsbingOLD 37973 Distribute proper substitution through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) Obsolete as of 18-Aug-2018. Use csbin 3865 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝐵𝐴 / 𝑥(𝐶𝐷) = (𝐴 / 𝑥𝐶𝐴 / 𝑥𝐷))
 
TheoremcsbresgOLD 37974 Distribute proper substitution through the restriction of a class. csbresgOLD 37974 is derived from the virtual deduction proof csbresgVD 38050. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbres 5211 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉𝐴 / 𝑥(𝐵𝐶) = (𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
 
TheoremcsbrngOLD 37975 Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 23-Aug-2018. Use csbrn 5404 instead. (New usage is discouraged.) (Proof modification is discouraged.)
(𝐴𝑉𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵)
 
Theoremcsbima12gALTOLD 37976 Move class substitution in and out of the image of a function. The proof is derived from the virtual deduction proof csbima12gALTVD 38052. (Contributed by Alan Sare, 10-Nov-2012.) Obsolete as of 20-Aug-2018. Use csbfv12 6025 instead. (Proof modification is discouraged.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐶𝐴 / 𝑥(𝐹𝐵) = (𝐴 / 𝑥𝐹𝐴 / 𝑥𝐵))
 
TheoremsspwtrALT2 37977 Short predicate calculus proof of the right-to-left implication of dftr4 4583. A class which is a subclass of its power class is transitive. This proof was constructed by applying Metamath's minimize command to the proof of sspwtrALT 37968, which is the virtual deduction proof sspwtr 37967 without virtual deductions. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)
 
TheorempwtrVD 37978 Virtual deduction proof of pwtr 4746; see pwtrrVD 37979 for the converse. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr 𝒫 𝐴)
 
TheorempwtrrVD 37979 Virtual deduction proof of pwtr 4746; see pwtrVD 37978 for the converse. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V       (Tr 𝒫 𝐴 → Tr 𝐴)
 
TheoremsuctrALT 37980 The successor of a transitive class is transitive. The proof of http://us.metamath.org/other/completeusersproof/suctrvd.html is a Virtual Deduction proof verified by automatically transforming it into the Metamath proof of suctrALT 37980 using completeusersproof, which is verified by the Metamath program. The proof of http://us.metamath.org/other/completeusersproof/suctrro.html is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. See suctr 5613 for the original proof. (Contributed by Alan Sare, 11-Apr-2009.) (Revised by Alan Sare, 12-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr suc 𝐴)
 
TheoremsnssiALTVD 37981 Virtual deduction proof of snssiALT 37982. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → {𝐴} ⊆ 𝐵)
 
TheoremsnssiALT 37982 If a class is an element of another class, then its singleton is a subclass of that other class. Alternate proof of snssi 4183. This theorem was automatically generated from snssiALTVD 37981 using a translation program. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → {𝐴} ⊆ 𝐵)
 
TheoremsnsslVD 37983 Virtual deduction proof of snssl 37984. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V       ({𝐴} ⊆ 𝐵𝐴𝐵)
 
Theoremsnssl 37984 If a singleton is a subclass of another class, then the singleton's element is an element of that other class. This theorem is the right-to-left implication of the biconditional snss 4162. The proof of this theorem was automatically generated from snsslVD 37983 using a tools command file, translateMWO.cmd, by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 ∈ V       ({𝐴} ⊆ 𝐵𝐴𝐵)
 
TheoremsnelpwrVD 37985 Virtual deduction proof of snelpwi 4738. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)
 
TheoremunipwrVD 37986 Virtual deduction proof of unipwr 37987. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 𝒫 𝐴
 
Theoremunipwr 37987 A class is a subclass of the union of its power class. This theorem is the right-to-left subclass lemma of unipw 4743. The proof of this theorem was automatically generated from unipwrVD 37986 using a tools command file , translateMWO.cmd , by translating the proof into its non-virtual deduction form and minimizing it. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴 𝒫 𝐴
 
TheoremsstrALT2VD 37988 Virtual deduction proof of sstrALT2 37989. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
TheoremsstrALT2 37989 Virtual deduction proof of sstr 3480, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 37988 using the command file translatewithout_overwriting.cmd . It was not minimized because the automated minimization excluding duplicates generates a minimized proof which, although not directly containing any duplicates, indirectly contains a duplicate. That is, the trace back of the minimized proof contains a duplicate. This is undesirable because some step(s) of the minimized proof use the proven theorem. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 
TheoremsuctrALT2VD 37990 Virtual deduction proof of suctrALT2 37991. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr suc 𝐴)
 
TheoremsuctrALT2 37991 Virtual deduction proof of suctr 5613. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 37990 using the tools command file translatewithout_overwritingminimize_excludingduplicates.cmd . (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(Tr 𝐴 → Tr suc 𝐴)
 
Theoremelex2VD 37992* Virtual deduction proof of elex2 3093. (Contributed by Alan Sare, 25-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ∃𝑥 𝑥𝐵)
 
Theoremelex22VD 37993* Virtual deduction proof of elex22 3094. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))
 
Theoremeqsbc3rVD 37994* Virtual deduction proof of eqsbc3r 3363. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → ([𝐴 / 𝑥]𝐶 = 𝑥𝐶 = 𝐴))
 
Theoremzfregs2VD 37995* Virtual deduction proof of zfregs2 8368. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ≠ ∅ → ¬ ∀𝑥𝐴𝑦(𝑦𝐴𝑦𝑥))
 
Theoremtpid3gVD 37996 Virtual deduction proof of tpid3g 4151. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
 
Theoremen3lplem1VD 37997* Virtual deduction proof of en3lplem1 8270. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 = 𝐴 → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
 
Theoremen3lplem2VD 37998* Virtual deduction proof of en3lplem2 8271. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝐴𝐵𝐵𝐶𝐶𝐴) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → ∃𝑦(𝑦 ∈ {𝐴, 𝐵, 𝐶} ∧ 𝑦𝑥)))
 
Theoremen3lpVD 37999 Virtual deduction proof of en3lp 8272. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ (𝐴𝐵𝐵𝐶𝐶𝐴)
 
20.30.7  Theorems proved using Virtual Deduction with mmj2 assistance
 
Theoremsimplbi2VD 38000 Virtual deduction proof of simplbi2 652. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
h1:: (𝜑 ↔ (𝜓𝜒))
3:1,?: e0a 37917 ((𝜓𝜒) → 𝜑)
qed:3,?: e0a 37917 (𝜓 → (𝜒𝜑))
The proof of simplbi2 652 was automatically derived from it. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 ↔ (𝜓𝜒))       (𝜓 → (𝜒𝜑))
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